Rank of a special matrix

Question

Say a $5\times 5$ matrix $$A = \left[ \begin{array}{ccc} 1&2&3&4&5\\ 6&7&8&9&10\\ 11&12&13&14&15\\ 16&17&18&19&20\\ 21&22&23&24&25\\ \end{array} \right] $$

or $n\times n$ in general, is there a quick way of finding the rank of this kind of matrix without using any row operations?

Thank you very much!

Answer 1

With this particularly special matrix, you can observe that if you look at a particular row $r$, then it is the average of row $r - i$ and row $r + i$ (provided $r - i \geq 1$ and $r + i \leq 5$, so that the rows we're talking about are actually there). More generally, though not necessary, is that row $a$ + row $d$ = row $b$ + row $c$ for $1 \leq a \leq b \leq c \leq d \leq 5$ such that $a + d = b + c$.

This tells you that all the intermediate rows are linear combinations of the first and last row, which are visibly linearly independent since neither is a multiple of the other. Hence, $\text{rank}(A) = 2$.